Finite Group Databasebeta

← #30639207

Subgroups — #30639207

Order 16256.

324 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861235C211yesmaximal, characteristic
2#22550642C211yesmaximal, characteristic, complemented
3#22549894C211yesmaximal, characteristic, complemented
4C2×#16861126C211yesmaximal, characteristic, complemented
5#22549646C211yesmaximal, characteristic, complemented
6C127⋊#338C211yesmaximal, characteristic, complemented
7#418×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—22no—
10#16861235—22no—
11C2×#14523236C412yescomplemented
12C2×#14523233C412yescomplemented
13#16861217—22no—
14#16861215—22no—
15C2×#14523232C2211yescharacteristic
16C2×#14523232C2211yescharacteristic
17C2×#14523232C2211yescharacteristic
18M16×D127—22no—
19#16861137—22no—
20#16861126—22no—
21C22×D4×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#102×C127—22no—
25#14523248—21no—
26#14523248—21no—
27#14523248D412yes—
28#14523248D412yes—
29C2×Dic254C2×C411yescharacteristic
30C2×D508C2×C411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33#14523236—24no—
34#14523233—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232—21no—
39#14523232—21no—
40#14523232D412yes—
41#14523231—24no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C2×C127⋊C8C2×C412yes—
45C8×D127—24no—
46C24×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127C2×C412yes—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56C2×C8×C127D412yes—
57C4⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C42×C127—21no—
60C22×D127—21no—
61Dic254—21no—
62Dic254C22⋊C412yes—
63D508—21no—
64D508C22⋊C412yes—
65C22⋊D127—22no—
66C22⋊D127—22no—
67C2×Dic127—21no—
68C4×D127—22no—
69C4×D127—22no—
70C127⋊C8—22no—
71C127⋊C8—22no—
72C127⋊C8—41no—
73C23×C127—22no—
74C23×C127—41no—
75C23×C127—41no—
76C23×C127—82no—
77C23×C127#4111yescharacteristic
78D4×C127—22no—
79D4×C127—22no—
80D4×C127—41no—
81D4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127C22×C411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C8×C127—22no—
90C22×C127—41no—
91C22×C127—42no—
92C22×C127—82no—
93C22×C127—82no—
94C22×C127—82no—
95C22×C127C4×D411yescharacteristic
96C22×C127#12111yescharacteristic
97C22×C127#12311yescharacteristic
98C2×D127—22no—
99C2×D127—81no—
100C4×C127—81no—
101C4×C127C2×C22⋊C411yescharacteristic
102C4×C127C4×D411yescharacteristic
103C4×C127#12011yescharacteristic
104C4×C127#12111yescharacteristic
105Dic127—22no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—21no—
109C2×C127—82no—
110C2×C127#38611yescharacteristic
111D127—42no—
112#1903—1271nomaximal, Sylow
113C127#190311yescharacteristic, complemented, Sylow
114C2×AGL(1,ℤ/8)—1271no—
115#568—1271no—
116#421—1271no—
117#418D12711yescharacteristic, complemented
118C2×#100—1271no—
119#344—1271no—
120#338—1271no—
121C2×#41—1271no—
122C22×D4C2×D12711yescharacteristic
123AGL(1,ℤ/8)—2542no—
124AGL(1,ℤ/8)—2542no—
125C2×SD16—1272no—
126C2×D8—1272no—
127#131—2542no—
128#131—2542no—
129C2×M16—1271no—
130C2×M16—1271no—
131C2×M16—1271no—
132C2×M16—2542no—
133C2×M16C2×D12711yescharacteristic
134#117C2×D12711yescharacteristic
135#104—2542no—
136#102—22no—
137#100—2542no—
138C24—21no—
139#41—1272no—
140#41—1272no—
141#41—2541no—
142#41—2541no—
143C2×Q8—1271no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—1271no—
150C2×D4—2541no—
151C2×D4C4×D12712yes—
152C22×C4—2541no—
153C22×C4C22×D12711yescharacteristic
154SD16—2544no—
155D8—2544no—
156M16—21no—
157M16—21no—
158M16—1272no—
159M16—2544no—
160M16—2541no—
161M16—2541no—
162M16—2541no—
163M16—2541no—
164M16—2541no—
165C2×C8—1272no—
166C2×C8—2544no—
167C2×C8—2541no—
168C2×C8—2541no—
169C2×C8C22⋊D12712yes—
170C4⋊C4—21no—
171C22⋊C4—21no—
172C42—21no—
173C23—22no—
174C23—41no—
175C23—41no—
176C23—82no—
177C23—2541no—
178C23#1452324711yescharacteristic
179Q8—1272no—
180Q8—2541no—
181D4—22no—
182D4—22no—
183D4—41no—
184D4—41no—
185D4—1272no—
186D4—2542no—
187D4—2542no—
188D4—2541no—
189C2×C4—41no—
190C2×C4—2541no—
191C2×C4—2542no—
192C2×C4—2542no—
193C2×C4C2×C4×D12711yescharacteristic
194C2×C4D4×D12711yescharacteristic
195C2×C4D4×D12711yescharacteristic
196C2×C4D4×D12711yescharacteristic
197C2×C4C2×C22⋊D12711yescharacteristic
198C2×C4#1452324811yescharacteristic
199C8—22no—
200C8—5081no—
201C8—2542no—
202C8—2542no—
203C22—41no—
204C22—42no—
205C22—82no—
206C22—82no—
207C22—82no—
208C22—10161no—
209C22—2542no—
210C22(C22⋊D127)⋊C411yescharacteristic
211C22#1686118011yescharacteristic
212C22#1686119111yescharacteristic
213C4—81no—
214C4—2542no—
215C4C22⋊C4×D12711yescharacteristic
216C4C4×C22⋊D12711yescharacteristic
217C4#1686117711yescharacteristic
218C4#1686118211yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—82no—
223C2—5082no—
224C2#2254975211yescharacteristic
subgroupG/H
centerC2#22549752
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#418×C127C2
socleC2×C127#386
prime radicalC2×#16861235C2
cosocleC22×C4×C127C23
hypercenter#418D127
hypocenterC127#1903
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22550642C2semidirect
3. #22549894C2semidirect
4. C2×#16861126C2semidirect
5. #22549646C2semidirect
6. C127⋊#338C2semidirect
7. #418×C127C2semidirect
11. C2×#14523236C4semidirect
12. C2×#14523233C4semidirect
113. C127#1903semidirect
117. #418D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168612352
#225506422
#225498942
C2×#168611262
#225496462
C127⋊#3382
#418×C1272
#1903127